Abstract : The mathematical analysis of optimization algorithms involves upper and lower bounds; we here focus on the second case. Whereas other chap- ters will consider black box complexity, we will here consider complexity based on the key assumption that the only information available on the ﬁtness values is the rank of individuals - we will not make use of the exact ﬁtness values. Such a reduced information is known eﬃcient in terms of ro- bustness Gelly et al., 2007, what gives a solid theoretical foundation to the robustness of evolution strategies, which is often argued without mathemat- ical rigor - and we here show the implications of this reduced information on convergence rates. In particular, our bounds are proved without inﬁ- nite dimension assumption, and they have been used since that time for designing algorithms with better performance in the parallel setting.